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Chapter 9: Problem 27

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1\) \(g(\pi)\)

Short Answer

Expert verified
The value of \(g(\pi)\) is \(-\pi^{2} + 4\pi + 1\).

Step by step solution

01

Identify the Function for Evaluation

Identify which function needs to be evaluated. In this case, it's the function \(g(x)\).
02

Substitute the Value

Substitute \(x = \pi\) into the function \(g(x) = -x^{2} + 4x + 1\).
03

Calculate the Squared Term

Evaluate \(\pi^{2}\). This gives us \(\pi^{2} \).
04

Apply Substitution to the Function

Substitute \(\pi\) and \(\pi^{2}\) into the function: \(g(\pi) = -\pi^{2} + 4\pi + 1\).
05

Simplify the Expression

Combine the terms: \[-\pi^{2} + 4\pi + 1\]. This is the final expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
Substitution is a fundamental concept in mathematics, where you replace a variable in an equation or expression with a specific value. This technique is especially useful when dealing with functions. For example, evaluating the function \( g(x) \) at \( x = \pi \) means substituting \( \pi \) for every instance of \( x \) in the function's expression.
In our example function \( g(x) = -x^{2} + 4x + 1 \), substituting \( x = \pi \) transforms it to \( g(\pi) = -\pi^{2} + 4\pi + 1 \).
Always double-check your substitutions to ensure accuracy, as a single mistake can lead to incorrect results.
Squared Term
Squared terms are expressions where a variable is raised to the power of two, such as \( x^{2} \). In function evaluation, these terms often require careful calculation to avoid errors. When you substitute a value into a squared term, you must square the substituted value.
For example, substituting \( x = \pi \) into \( x^{2} \) gives \( \pi^{2} \). This term is crucial because it impacts the overall value of the function.
Remember, squaring a number means multiplying it by itself: \( \pi \times \pi = \pi^{2} \).
Accurate computation of squared terms ensures the entire function evaluation remains correct.
Simplify Expression
Simplifying an expression means combining like terms and performing basic arithmetic to make the expression more manageable. After substituting values into a function, your next step is to simplify the resulting expression.
Using our example, after substituting \( x = \pi \) into the function \( g(x) = -x^{2} + 4x + 1 \), you get \( -\pi^{2} + 4\pi + 1 \).
To simplify, you just combine the individual terms:
  • \( -\pi^{2} \)
  • \( 4\pi \)
  • \( +1 \)
This results in the final expression, which is already simplified in this case: \( -\pi^{2} + 4\pi + 1 \).
Simplification is a key step, ensuring you present the most reduced form of the expression for clarity and further use.

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